Existence of Inverse Elementary Column Operation/Exchange Columns

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Theorem

Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$.

Let $\mathbf A \in \map \MM {m, n}$ be a matrix.


Let $\map e {\mathbf A}$ be the elementary column operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$.

\((\text {ECO} 3)\)   $:$   \(\ds \kappa_k \leftrightarrow \kappa_l \)    Interchange columns $k$ and $l$      


Let $\map {e'} {\mathbf A'}$ be the inverse of $e$.


Then $e'$ is the elementary column operation:

$e' := \kappa_k \leftrightarrow \kappa_l$

That is:

$e' = e$


Proof

In the below, let:

$\kappa_k$ denote column $k$ of $\mathbf A$
$\kappa'_k$ denote column $k$ of $\mathbf A'$
$\kappa_k$ denote column $k$ of $\mathbf A$

for arbitrary $k$ such that $1 \le k \le n$.


By definition of elementary column operation:

only the column or columns directly operated on by $e$ is or are different between $\mathbf A$ and $\mathbf A'$

and similarly:

only the column or columns directly operated on by $e'$ is or are different between $\mathbf A'$ and $\mathbf A$.

Hence it is understood that in the following, only those columns directly affected will be under consideration when showing that $\mathbf A = \mathbf A$.


Let $\map e {\mathbf A}$ be the elementary column operation:

$e := \kappa_k \leftrightarrow \kappa_l$

Thus we have:

\(\ds \kappa'_k\) \(=\) \(\ds \kappa_l\)
\(\, \ds \text {and} \, \) \(\ds \kappa'_l\) \(=\) \(\ds \kappa_k\)


Now let $\map {e'} {\mathbf A'}$ be the elementary column operation which transforms $\mathbf A'$ to $\mathbf A$:

$e' := \kappa'_k \leftrightarrow \kappa'_l$


Applying $e'$ to $\mathbf A'$ we get:

\(\ds \kappa_k\) \(=\) \(\ds \kappa'_l\)
\(\, \ds \text {and} \, \) \(\ds \kappa_l\) \(=\) \(\ds \kappa'_k\)
\(\ds \leadsto \ \ \) \(\ds \kappa_k\) \(=\) \(\ds \kappa_k\)
\(\, \ds \text {and} \, \) \(\ds \kappa_l\) \(=\) \(\ds \kappa_l\)
\(\ds \leadsto \ \ \) \(\ds \mathbf A\) \(=\) \(\ds \mathbf A\)

It is noted that for $e'$ to be an elementary column operation, the only possibility is for it to be as defined.

$\blacksquare$

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